The Unapologetic Mathematician

Mathematics for the interested outsider

Product Measures

We continue as yesterday, considering the two \sigma-finite measure spaces (X,\mathcal{S},\mu) and (Y,\mathcal{T},\nu), and the product measure space (X\times Y,\mathcal{S}\times\mathcal{T}).

Last time we too a measurable set E\subseteq X\times Y and defined the functions f(x)=\nu(E_x) and g(y)=\mu(E^y). We also showed that

\displaystyle\int f\,d\mu=\int g\,d\nu

That is, for every measurable E we can define the real number

\displaystyle\lambda(E)=\int\nu(E_x)\,d\mu=\int\mu(E^y)\,d\nu

I say that this function \lambda is itself a \sigma-finite measure, and that for any measurable rectangle A\times B we have \lambda(A\times B)=\mu(A)\nu(B). Since measurable rectangles generate the \sigma-ring \mathcal{S}\times\mathcal{T}, this latter condition specifies \lambda uniquely.

To see that \lambda is a measure, we must show that it is countably additive. If \{E_n\} is a sequence of disjoint sets then we calculate

\displaystyle\begin{aligned}\lambda\left(\biguplus\limits_{n=1}^\infty E_n\right)&=\int\nu\left(\left(\biguplus\limits_{n=1}^\infty E_n\right)_x\right)\,d\mu\\&=\int\nu\left(\biguplus\limits_{n=1}^\infty(E_n)_x\right)\,d\mu\\&=\int\sum\limits_{n=1}^\infty\nu\left((E_n)_x\right)\,d\mu\\&=\sum\limits_{n=1}^\infty\int\nu\left((E_n)_x\right)\,d\mu\\&=\sum\limits_{n=1}^\infty\lambda(E_n)\end{aligned}

where we have used the monotone convergence theorem to exchange the sum and the integral.

We verify the \sigma-finiteness of \lambda by covering each measurable set E by countably many measurable rectangles with finite-measure sides. Since the sides’ measures are finite, the measure of the rectangle itself is the product of two finite numbers, and is thus finite.

We call the measure \lambda the “product” of the measures \mu and \nu, and we write \lambda=\mu\times\nu. We thus have a \sigma-finite measure space (X\times Y,\mathcal{S}\times\mathcal{T},\mu\times\nu) that we call the “cartesian product” of the spaces (X,\mathcal{S},\mu) and (Y,\mathcal{T},\nu).

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July 23, 2010 - Posted by | Analysis, Measure Theory

6 Comments »

  1. Professor Armstrong,

    I think that Feynman’s path integrals of quantum field theory might provide the kind of product measures that we were discussing although I am not so sure how rigorous they are considered to be mathemnatically. I wished to email this note to you, but I could not figure out where to find your email. It is probably not posted.

    Comment by Hamid | July 25, 2010 | Reply

  2. They’re completely unrigorous, mathematically; that’s the whole issue with them! The path-integral is a wonderful heuristic, that can be interpreted in many cases to give a mathematically-sensible statement.

    But in general there is no known coherent analogue of something like Lebesgue measure on function spaces — no way to “integrate over all paths” as the plain reading of the path-integral formulation suggests. Finding a universally-applicable interpretation of the path-integral heuristic instead of problem-specific ad hoc methods is the single greatest puzzle of mathematical physics.

    Comment by John Armstrong | July 25, 2010 | Reply

  3. [...] of . Now if we have a measure on and Lebesgue measure on the Borel sets, we can define the product measure on . Since we know and are both measurable, we can investigate their measures. I assert [...]

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  4. [...] and Iterated Integrals Let and be two -finite measure spaces, and let be the product measure on [...]

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  5. [...] We continue our assumptions that and are both -finite measure spaces, and we consider the product space [...]

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  6. I can suggest to see the following papers where can be found constructions of partial analogs of Lebesgue measure on some function spaces. There are

    1) Baker, R., “Lebesgue measure” on $R\sp \infty$. Proc. Amer.
    Math. Soc., 113(4) (1991), 1023–1029.

    2) Baker, R., “Lebesgue measure” on $\Bbb R\sp \infty$. II.Proc. Amer. Math. Soc., 132(9), (2004), 2577–2591

    3) Pantsulaia,G., On ordinary and Standard Lebesgue Measures on $R^{\infty}$, Bull. Polish Acad. Sci.73(3) (2009), 209-222.

    4) Pantsulaia,G., On a standard product of an arbitrary family of sigma-finite Borel measures with domain in Polish spaces, Theory Stoch. Process, vol. 16(32), 2010, no 1, p.84-93.

    Comment by Gogi Pantsulaia | December 17, 2010 | Reply


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